Chapter 4 The debroglie hypothesis

Transcription

1 Capter 4 Te debrglie yptesis In 194, te Frenc pysicist Luis de Brglie after lking deeply int te special tery f relatiity and ptn yptesis,suggested tat tere was a mre fundamental relatin between waes and particles, tus accrding t te special tery f relatiity fr a ptn, E and its mmentum Tese tw sets f quantities are cnnected trug plank Luis debrglie suggested tat te prblem is symmetrical and tat te dual (wae and particle) nature f electrmagnetic radiatin may be extended t material particles suc as electrns, prtns, neutrns, etc. te difference being te ptn ae m =0 and =C,wile te material particle,ae m 0 0 and m<c. Te abe type f argument led debrglie t suggest te fllwing yptesis applicable t all matter as well as particles. a) if tere is a particle f mmentum its mtin is assciated wit (r guided by ) a wae f waelengt b) if tere is a wae f waelengt, te square f te amplitude f te wae at any pint in space, a particle f mmentum Te implicatin is tat depending n te circumstances, a wae may be bsered as a particle, wile particle may be bsered as a wae. 38

2 Example Capter 4 Calculate te debrglie waelengt assciated wit te fllwing a) A glf ball f mass 50 g ming wit a elcity f 0 m/s. b) A prtn ming wit a elcity f 00 m/s. c) An electrn ming wit a kinetic energy f 10 ev. 39

3 Te diffractin f waes Capter 4 Te spatial in intensity (wit alternating maxima and minima) bsered after a beam f ligt as passed trug a slit is called diffractin. Frm ur study f te general pysics, we ae seen tat mecanical diffractin gratings aing as many as 100 line per centimeter t prduce diffractin f isible ligt. if te beam is incident nrmally t te grating,te cnditin fr diffractin maxima is gien by d sin n Were n=1,,3,.. is te rder f diffractin maximum. Diffractin f particles In rder t cnfirm te De-Brglie yptesis, we must sw tat particles suc as electrns,prtns,neutrns.atms and ters exibits diffractin patterns similar t tse f x-rays (r any ter electrmagnetic radiatin) t get sme idea. Let us calculate te De- Brglie waelengt assciated wit an electrn wic as been accelerated trug a ptential difference f V. let m be te mass f te electrn and te elcity f suc an electrn, ten nnrelatiisitically 1 m ev ev m If we use tis alue, te debrglie waelengt is gien by Tat is m mev x9.108x x x1.6 x10 19 xv 150 x10 V cm 40

4 Or if V is expressed in lt 150 V Capter 4 Because f te speed f particles is always less tan te speed f ligt. S te speed f De-Brglie wae assciated is always mre tan C. Example re tat tere is n rest ptn in nature We knw tat m Ttal energy = E mc S E c (1) c E Wen we use te relatin f relatiistic mass m m 1 / c By squaring we get m () were m is te rest mass f a particle 1 / c Using (1) in () c m ( ) E c 1 ( ) / c E m E c 4 c ande f m ( c c W f f 1 ) 1 / 41

5 Capter 4 1/ S W 1 m ( ) c c m W 4 c c W c m c c ) c ( mc W c ( 1) Let us suppse W c We get m ( c 1) 1 m c ( m 0 0 S tere is n rest mass fr ptn. Example re tat te debrglie speed wae is gien by W c / It is nt imprtant fr te debrglie speed wae assciated wit a particle t be te same as te particle speed. If equal t te De-Brglie waelengt and f its frequency ten te speed wae W f...(1) By using plank yptesis E f f E / 4

6 S equatin (1) becmes E W...() By using debrglie yptesis S we can write () E W...(3) W mc m c Capter 4 / We knw tat E=mc and p=m, substitute tese alues in (3) we get Te debrglie wae grup Te pase elcity W k Wile te elcity u f te wae grup d Grup elcity u dk Depending n te manner in wic pase elcity aries wit wae number in a particle medium, te grup elcity may be greater r less tan te pase elcity, if te pase elcity W is te same fr all waelengts, as is true fr ligt waes in acuum, te grup elcity and pase elcities are te same. Te angular frequency and wae number f te de-brglie waes assciated wit a bdy at rest mass m ming wit te elcity are. mc m k m 1 m c 1 / c / c Bt and k are functin f te elcity. 43

7 Capter 4 Te pase elcity W is as we fund earlier. c W k Wic exceeds bt te elcity f te bdy and te elcity f ligt c since <c Te grup elcity u f te de-brglie waes assciated wit te bdy is, d d / d u dk dk / d d m d (1 / c ) 3 / S u Tis means tat te debrglie wae grup assciated wit a ming bdy traels wit te same elcity as te bdy. Te Heisenberg uncertainty principle Let us take an experiment in wic we try t measure te psitin f an electrn. Object can be seen t accuracy n greater tan te waelengt f te radiatin used. If we want accurate psitin measurement, we must use a srt waelengt,but a srt waelengt crrespnds t ig frequency and ig energy,and mre energy te ptns ae,te lnger waelengt, and crrespndingly lwer energy are used, te bject's mtin wen struck by te ptns will nt be affected as muc, but its psitin will be less accurately knwn. Tus te act f bsering prduces a significant uncertainty in eiter te psitin r te mmentum f te electrn. We can measure at te best t an accuracy f abut. Tat is, te uncertainty in te psitin measurement x is apprximately x Suppse tat te bject can be detected by a single ptn, te ptn as a mmentum / and wen it strikes ur bject it will gie sme r 44

8 Capter 4 f all f tis mmentum t te bject. Terefre te final mmentum f ur bject will be uncertain in te amunt Since we can't tell befreand w muc mmentum will be transferred? te prduct f tese uncertainties ( x)( ) Of curse, te uncertainties culd be wrse tan tis, depending n te apparatus and te number f ptns needed fr detectin. In Heisenberg's mre careful calculatin, e fund tat at te ery best ( x)( ) Tis is a matematical statement f Heisenberg's uncertainty principle. It tells us tat we can nt measure bt te psitin and mmentum f an bject precisely at te same time. Te mre accurately we try t measure te psitin s tat x is small, te greater will be te uncertainty in mmentum. Example An electrn initially at rest is accelerated trug a ptential difference f 54 lts cmpute 1) te elcity f te electrn ) debrglie waelengt 3) pase elcity f te electrn wae 45

9 Capter 4 Example Wat is te De-Brglie waelengt f neutrn wse energy is 1 ev. 46

10 Capter 4 Hmewrk 1- Wat is te De-Brglie waelengt f an electrn accelerated trug 0000 lt? - An electrn initially at rest is accelerated trug a ptential difference f 5000 lt. Cmpute i) te mmentum ii) te debrglie waelengt iii) te wae number f te electrn wae 3- Cmpute te debrglie waelengt f a prtn wse kinetic energy is equal t te rest energy f an electrn. Mass f a prtn is 1837 times tat f te electrn. 4- Calculate te De-Brglie waelengt assciated wit an electrn wit a kinetic energy f 510 KeV. 5- Wat suld be te kinetic energy f a prtn s tat te debrglie waelengt assciated wit is 5000 Ǻ? 6- Trug wat ptential difference suld an electrn be accelerated s tat te De-Brglie waelengt assciated wit it is 0.1? 47